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On the Global Strong Solutions to Magnetohydrodynamics with Density-Dependent Viscosity and Degenerate Heat-Conductivity in Unbounded Domains

Yuebo Cao, Yi Peng, Ying Sun

Abstract

For the equations of a planar magnetohydrodynamic (MHD) compressible flow with the viscosity depending on the specific volume of the gas and the heat conductivity being proportional to a positive power of the temperature, we obtain global existence of the unique strong solutions to the Cauchy problem or the initial-boundary-value one under natural conditions on the initial data in one-dimensional unbounded domains. Our result generalizes the classical one of the compressible Navier-Stokes system with constant viscosity and heat conductivity ([Kazhikhov. Siberian Math. J. (1982)]) to the planar MHD compressible flow with nonlinear viscosity and degenerate heat-conductivity, which means no shock wave, vacuum, or mass or heat concentration will be developed in finite time, although the interaction between the magnetodynamic effects and hydrodynamic is complex and the motion of the flow has large oscillations.

On the Global Strong Solutions to Magnetohydrodynamics with Density-Dependent Viscosity and Degenerate Heat-Conductivity in Unbounded Domains

Abstract

For the equations of a planar magnetohydrodynamic (MHD) compressible flow with the viscosity depending on the specific volume of the gas and the heat conductivity being proportional to a positive power of the temperature, we obtain global existence of the unique strong solutions to the Cauchy problem or the initial-boundary-value one under natural conditions on the initial data in one-dimensional unbounded domains. Our result generalizes the classical one of the compressible Navier-Stokes system with constant viscosity and heat conductivity ([Kazhikhov. Siberian Math. J. (1982)]) to the planar MHD compressible flow with nonlinear viscosity and degenerate heat-conductivity, which means no shock wave, vacuum, or mass or heat concentration will be developed in finite time, although the interaction between the magnetodynamic effects and hydrodynamic is complex and the motion of the flow has large oscillations.

Paper Structure

This paper contains 2 sections, 9 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm1.1']}

Key Result

Theorem \oldthetheorem

Suppose that and that the initial data $( v_0,u_0,\theta_0,{\bold b}_0,{\bold w}_0)$ satisfies and and are compatible with 1.10, 1.11. Then there exists a unique global strong solution $(v,u,\theta,{\bold b},{\bold w})$ to the initial-boundary-value problem 1.1-1.9, or 1.1-1.81.10, or 1.1-1.81.11 satisfying for any $T>0$, and for each $(x,t)\in \Omega \times[0,T]$ where $C>0$ is a constant dep

Theorems & Definitions (11)

  • Theorem \oldthetheorem
  • Remark 1.1
  • Remark 1.2
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • ...and 1 more